I'd say the main difference is that OAuth is granting the SP the ability to "do stuff" as the original user (including reading the user's profile details, as OIDC does), as opposed to SAML's approach of just sending attributes describing them.
For what it's worth, it is certainly possible for SAML SPs to flag that certain attributes should/must be released to them via their metadata, but the actual release is at the whim of the IdP and its operators. It's also possible for a SAML IdP to expose that level of detail to its end users and allow them to agree/disagree to the attribute release, although I'd be surprised if that behaviour was particularly common in practice.
The difference between OIDC and OAuth boils down to exchanging attribute assertions describing a user as opposed to the delegation of a specific set of allowed actions, as OAuth was intended to do. OIDC and SAML are basically the same thing, with OIDC being a somewhat less frightening and more modern protocol.
Reading the user's profile information _is_ the delegated action. OAuth providers were already doing this prior to OIDC but in incompatible ways. OIDC standardized how that information is requested and returned.
For what it's worth, it is certainly possible for SAML SPs to flag that certain attributes should/must be released to them via their metadata, but the actual release is at the whim of the IdP and its operators. It's also possible for a SAML IdP to expose that level of detail to its end users and allow them to agree/disagree to the attribute release, although I'd be surprised if that behaviour was particularly common in practice.